What does the "number of channels" correspond to in U-Net?
In this example you have a gray scale image of size 572x572 and 1 (gray) channel. The first convolution operation consists of 64 filters of size 3x3 and 1 channel per filter. The channel of the filters always fits the channel size of the previous layer (here: the Input).
In the second convolution step of this explicit architecture, you again use 64 filters of size 3x3. In this case, each of these filters consists of 64 channels according to the previous output (64 feature maps/channels). The output of the second convolution consists of 64 feature maps according to the amount of 64 filters in the second convolution. This video from Andrew Ng visualizes it perfectly.